Hey I am also new and I have a similar problem to you I think, but I might be able to shed some light on to your problem.
I assume that you have tried changing the batteries in the fob? I know its simple, but we may aswell eliminate the simple things first. The LED requires very little power to light up, so it is very possible that the battery can power this function, but does not have enough juice to unlock/disarm your car alarm.
Like I said before, I am having a similar problem to you, and I have a feeling that it might not even be the fob which is the problem. My knowledge on remote locking systems is minimal, but there must be a sensor within the car which receives the signal from the key. I believe this is probably where my problem is coming from.
Try waving your key just above the windscreen and pressing the unlock button a few times, mine will open this way for some reason. (perhaps the internal car sensor is very close to this area?)
As for opening your car with the key...
From what I have read (This is totally untested on my part, so please don't blame me if it doesn't work), you can open your car with the key and the alarm will go off. Apparently, if you then start the engine, it will stop the alarm.
Good luck :P
I hope this helps you anyway, please keep me informed on your progress because I would really like to know of any fixes people come up with.